Finite Group Databasebeta

← #13159281

Subgroups — #13159281

Order 1728.

720 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64096C211yesmaximal, characteristic
2#11723535C211yesmaximal, characteristic
3#11722178C211yesmaximal, characteristic
4C32×#808C211yesmaximal, characteristic
5#11722051C211yesmaximal, characteristic
6#11722040C211yesmaximal, characteristic
7#11722039C211yesmaximal, characteristic
8#10592553—31nomaximal
9#10590556—34nomaximal
10C2×C3×#634C2211yescharacteristic
11C2×#6293—21no—
12C33×C4⋊C4C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×#3750—21no—
16C3×C4×C3⋊Dic3—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88807—21no—
21#64097—31no—
22#64096D311yescharacteristic
23C3×#887—34no—
24#63918—34no—
25C32×#106—31no—
26#63643—31no—
27#63641—31no—
28#63637—31no—
29#63635—31no—
30#63635—34no—
31#63635—64no—
32C3×#808D314yes—
33Dic3⋊(C3⋊C8)—34no—
34#63566—34no—
35#63565—34no—
36C2×C33×C4—21no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39C2×C3×C3⋊Dic3—21no—
40C2×C3×C3⋊Dic3D411yescharacteristic
41C2×C3×C3⋊Dic3D411yescharacteristic
42C3×#634—21no—
43C3×#634Q812yescomplemented
44C2×C32×Dic3—21no—
45#6293—42no—
46C3⋊(C3⋊(C3⋊C8))—21no—
47C32×C3⋊C8—21no—
48#5020—91no—
49#4854—94no—
50C2×#634—61no—
51C2×#634C2×D311yescharacteristic
52C2×C3×Dic6—34no—
53C2×#627—64no—
54C2×C32×C8—31no—
55C32×C4⋊C4—24no—
56C32×C4⋊C4—31no—
57C32×C4⋊C4C2×D314yes—
58C32×C4⋊C4C2×D311yescharacteristic
59C4⋊(C3⋊Dic3)—31no—
60C4⋊(C3⋊Dic3)—34no—
61C4⋊(C3⋊Dic3)—64no—
62#3750—21no—
63#3750—61no—
64C4×C3⋊Dic3—21no—
65C4×C3⋊Dic3C2×D311yescharacteristic
66C2×C3⋊(C3⋊C8)—31no—
67C2×C3⋊(C3⋊C8)—34no—
68C2×C3⋊(C3⋊C8)—64no—
69C3×C4⋊Dic3C2×D314yes—
70C3×Dic3⋊C4—64no—
71C3×C4×Dic3—34no—
72C3×C4×Dic3—64no—
73C2×C3×C3⋊C8C2×D314yes—
74Dic3⋊Dic3—64no—
75C22×C33C2×D411yescharacteristic
76C33×C4—41no—
77C33×C4C2×Q811yescharacteristic
78C33×C4#4111yescharacteristic
79C3⋊(C3⋊Dic3)—41no—
80C3×C3⋊Dic3—22no—
81C3×C3⋊Dic3—21no—
82C3×C3⋊Dic3SD1612yes—
83C32×Dic3—41no—
84C3×C4×Q8—91no—
85#888—94no—
86#887—34no—
87#869—91no—
88C3×#106—34no—
89#835—91no—
90#835—91no—
91#816—94no—
92Dic3⋊C8—94no—
93#810—94no—
94#808—94no—
95#808—91no—
96#808—184no—
97#808C3⋊D311yescharacteristic
98C4⋊(C3⋊C8)—91no—
99C2×C32×C4—21no—
100C2×C32×C4—24no—
101C2×C32×C4—24no—
102C2×C32×C4—44no—
103C2×C32×C4—61no—
104C2×C32×C4C22×D314yes—
105C2×C32×C4C22×D311yescharacteristic
106C2×C3⋊Dic3—21no—
107C2×C3⋊Dic3—61no—
108C2×C3⋊Dic3—64no—
109C2×C3⋊Dic3—124no—
110C2×C3⋊Dic3#6711yescharacteristic
111C2×C3⋊Dic3#6711yescharacteristic
112#634—21no—
113#634—122no—
114#634Dic612yescomplemented
115C2×C3×Dic3—24no—
116C2×C3×Dic3—34no—
117C2×C3×Dic3—34no—
118C2×C3×Dic3—64no—
119C3×Dic6—38no—
120C3×Dic6—64no—
121#627—128no—
122C32×C8—61no—
123C3⋊(C3⋊C8)—68no—
124C3⋊(C3⋊C8)—64no—
125C3⋊(C3⋊C8)—61no—
126C3×C3⋊C8—24no—
127#475—271noSylow
128C2×C33C22⋊Q811yescharacteristic
129C2×C33C2×SD1611yescharacteristic
130C2×C33#13711yescharacteristic
131C2×C3×Q8—91no—
132C2×Dic6—34no—
133C2×Dic6—181no—
134C2×Dic6—184no—
135C2×C3×C8—34no—
136C3×C4⋊C4—24no—
137C3×C4⋊C4—34no—
138C3×C4⋊C4—181no—
139C3×C4⋊C4D3214yes—
140C3×C4⋊C4C2×C3⋊D311yescharacteristic
141C3×C42—91no—
142C3×C42—181no—
143C4⋊Dic3—94no—
144C4⋊Dic3—91no—
145C4⋊Dic3—184no—
146C4⋊Dic3—181no—
147C4⋊Dic3C2×C3⋊D311yescharacteristic
148Dic3⋊C4—64no—
149Dic3⋊C4—184no—
150C4×Dic3—34no—
151C4×Dic3—64no—
152C2×C3⋊C8—94no—
153C2×C3⋊C8—91no—
154C2×C3⋊C8—184no—
155C2×C3⋊C8C2×C3⋊D311yescharacteristic
156C22×C32—24no—
157C22×C32D3×D414yes—
158C22×C32C2×#6711yescharacteristic
159C32×C4—24no—
160C32×C4—24no—
161C32×C4—41no—
162C32×C4—44no—
163C32×C4—84no—
164C32×C4—121no—
165C32×C4C2×Dic611yescharacteristic
166C32×C4#23514yes—
167C32×C4#23511yescharacteristic
168C32×C4D3×Q814yes—
169C3⋊Dic3—22no—
170C3⋊Dic3—21no—
171C3⋊Dic3—121no—
172C3⋊Dic3—124no—
173C3⋊Dic3—244no—
174C3⋊Dic3#21512yes—
175C3×Dic3—38no—
176C3×Dic3—44no—
177C3×Dic3—68no—
178C3×Dic3—64no—
179C4⋊Q8—271no—
180C4×Q8—91no—
181#106—31no—
182#106—271no—
183C4⋊C8—271no—
184#103—271no—
185#103—271no—
186C33#47511yescharacteristic, complemented, Sylow
187C3×Q8—92no—
188C3×Q8—181no—
189C2×C3×C4—24no—
190C2×C3×C4—21no—
191C2×C3×C4—24no—
192C2×C3×C4—44no—
193C2×C3×C4—64no—
194C2×C3×C4—91no—
195C2×C3×C4—91no—
196C2×C3×C4—181no—
197C2×C3×C4C2×D3214yes—
198C2×C3×C4C22×C3⋊D311yescharacteristic
199C2×Dic3—21no—
200C2×Dic3—34no—
201C2×Dic3—34no—
202C2×Dic3—64no—
203C2×Dic3—184no—
204C2×Dic3—181no—
205C2×Dic3—364no—
206Dic6—38no—
207Dic6—64no—
208Dic6—368no—
209Dic6—362no—
210C3×C8—64no—
211C3⋊C8—21no—
212C3⋊C8—188no—
213C3⋊C8—184no—
214C3⋊C8—181no—
215C2×C32—24no—
216C2×C32—24no—
217C2×C32—24no—
218C2×C32#89614yes—
219C2×C32D3×SD1614yes—
220C2×C32#91514yes—
221C2×C32#92411yescharacteristic
222C2×C32C2×#21511yescharacteristic
223C2×C32#95111yescharacteristic
224C2×Q8—91no—
225C2×Q8—541no—
226C2×C8—31no—
227C2×C8—271no—
228C4⋊C4—31no—
229C4⋊C4—181no—
230C4⋊C4—271no—
231C4⋊C4—541no—
232C4⋊C4D3×C3⋊D311yescharacteristic
233C42—91no—
234C42—181no—
235C22×C3—24no—
236C22×C3D3×#6714yes—
237C22×C3D4×C3⋊D311yescharacteristic
238C3×C4—24no—
239C3×C4—24no—
240C3×C4—41no—
241C3×C4—44no—
242C3×C4—84no—
243C3×C4—92no—
244C3×C4—124no—
245C3×C4—182no—
246C3×C4—181no—
247C3×C4D3×Dic614yes—
248C3×C4#379814yes—
249C3×C4#382811yescharacteristic
250C3×C4Q8×C3⋊D311yescharacteristic
251Dic3—38no—
252Dic3—41no—
253Dic3—68no—
254Dic3—64no—
255Dic3—361no—
256Dic3—364no—
257Dic3—724no—
258C32—24no—
259C32#485414yescomplemented
260C32#502011yescharacteristic, complemented
261Q8—92no—
262Q8—181no—
263Q8—1082no—
264C2×C4—21no—
265C2×C4—61no—
266C2×C4—91no—
267C2×C4—91no—
268C2×C4—181no—
269C2×C4—541no—
270C2×C4C2×D3×C3⋊D311yescharacteristic
271C8—61no—
272C8—541no—
273C2×C3—24no—
274C2×C3—24no—
275C2×C3—24no—
276C2×C3#6385114yes—
277C2×C3#6392914yes—
278C2×C3D3×#21514yes—
279C2×C3#6410511yescharacteristic
280C2×C3SD16×C3⋊D311yescharacteristic
281C2×C3#6412411yescharacteristic
282C22C3⋊D3×#6711yescharacteristic
283C4—41no—
284C4—92no—
285C4—121no—
286C4—181no—
287C4—182no—
288C4—1081no—
289C4C3⋊D3×Dic611yescharacteristic
290C4#8902011yescharacteristic
291C3—24no—
292C3#1059055614yescomplemented
293C3#1059255311yescharacteristic, complemented
294C2#1172341811yescharacteristic
295C2#1172355211yescharacteristic
296C2C3⋊D3×#21511yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#475
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
43. C3×#634Q8semidirect
114. #634Dic6semidirect
186. C33#475semidirect
259. C32#4854semidirect
260. C32#5020semidirect
292. C3#10590556semidirect
293. C3#10592553semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×#640962
#117235352
#117221782
C32×#8082
#117220512
#117220402
#117220392
#105925533
#105905563